Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Pockels effect

Peter Hertel

University of Osnabruck, Germany

Lecture presented at APS, Nankai University, China

http://www.home.uni-osnabrueck.de/phertel

October/November 2011

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Overview

• optical medium in an external electric field

• symmetry considerations

• lithium niobate

• electro-optic devices

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Overview

• optical medium in an external electric field

• symmetry considerations

• lithium niobate

• electro-optic devices

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Overview

• optical medium in an external electric field

• symmetry considerations

• lithium niobate

• electro-optic devices

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Overview

• optical medium in an external electric field

• symmetry considerations

• lithium niobate

• electro-optic devices

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Overview

• optical medium in an external electric field

• symmetry considerations

• lithium niobate

• electro-optic devices

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

External electric field

• Optical properties depend on the equilibrium state ofmatter

• Temperature, pressure or strain, external fields, . . .

• εij(ω;T, S,E,B, . . . )• here εij(ω;E)

• linear electrooptics

ε(ω,E)−1ij = ε(ω, 0)−1

ij + rijkEk + . . .

• Pockels coefficients rijk form tensor of rank 3

• not all crystals can have such tensors

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

External electric field

• Optical properties depend on the equilibrium state ofmatter

• Temperature, pressure or strain, external fields, . . .

• εij(ω;T, S,E,B, . . . )• here εij(ω;E)

• linear electrooptics

ε(ω,E)−1ij = ε(ω, 0)−1

ij + rijkEk + . . .

• Pockels coefficients rijk form tensor of rank 3

• not all crystals can have such tensors

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

External electric field

• Optical properties depend on the equilibrium state ofmatter

• Temperature, pressure or strain, external fields, . . .

• εij(ω;T, S,E,B, . . . )• here εij(ω;E)

• linear electrooptics

ε(ω,E)−1ij = ε(ω, 0)−1

ij + rijkEk + . . .

• Pockels coefficients rijk form tensor of rank 3

• not all crystals can have such tensors

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

External electric field

• Optical properties depend on the equilibrium state ofmatter

• Temperature, pressure or strain, external fields, . . .

• εij(ω;T, S,E,B, . . . )

• here εij(ω;E)

• linear electrooptics

ε(ω,E)−1ij = ε(ω, 0)−1

ij + rijkEk + . . .

• Pockels coefficients rijk form tensor of rank 3

• not all crystals can have such tensors

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

External electric field

• Optical properties depend on the equilibrium state ofmatter

• Temperature, pressure or strain, external fields, . . .

• εij(ω;T, S,E,B, . . . )• here εij(ω;E)

• linear electrooptics

ε(ω,E)−1ij = ε(ω, 0)−1

ij + rijkEk + . . .

• Pockels coefficients rijk form tensor of rank 3

• not all crystals can have such tensors

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

External electric field

• Optical properties depend on the equilibrium state ofmatter

• Temperature, pressure or strain, external fields, . . .

• εij(ω;T, S,E,B, . . . )• here εij(ω;E)

• linear electrooptics

ε(ω,E)−1ij = ε(ω, 0)−1

ij + rijkEk + . . .

• Pockels coefficients rijk form tensor of rank 3

• not all crystals can have such tensors

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

External electric field

• Optical properties depend on the equilibrium state ofmatter

• Temperature, pressure or strain, external fields, . . .

• εij(ω;T, S,E,B, . . . )• here εij(ω;E)

• linear electrooptics

ε(ω,E)−1ij = ε(ω, 0)−1

ij + rijkEk + . . .

• Pockels coefficients rijk form tensor of rank 3

• not all crystals can have such tensors

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

External electric field

• Optical properties depend on the equilibrium state ofmatter

• Temperature, pressure or strain, external fields, . . .

• εij(ω;T, S,E,B, . . . )• here εij(ω;E)

• linear electrooptics

ε(ω,E)−1ij = ε(ω, 0)−1

ij + rijkEk + . . .

• Pockels coefficients rijk form tensor of rank 3

• not all crystals can have such tensors

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Non-centrosymmetric crystals

• ε(ω,E)ij must be real and symmetric

• rijk also must be real and symmetric in the first index pair

• If crystal has an inversion center, then space inversionbrings one minus sign per index

• hence rijk must vanish

• only crystals without inversion center can show the

Pockels effect

• lithium niobate with 3m symmetry is an example

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Non-centrosymmetric crystals

• ε(ω,E)ij must be real and symmetric

• rijk also must be real and symmetric in the first index pair

• If crystal has an inversion center, then space inversionbrings one minus sign per index

• hence rijk must vanish

• only crystals without inversion center can show the

Pockels effect

• lithium niobate with 3m symmetry is an example

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Non-centrosymmetric crystals

• ε(ω,E)ij must be real and symmetric

• rijk also must be real and symmetric in the first index pair

• If crystal has an inversion center, then space inversionbrings one minus sign per index

• hence rijk must vanish

• only crystals without inversion center can show the

Pockels effect

• lithium niobate with 3m symmetry is an example

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Non-centrosymmetric crystals

• ε(ω,E)ij must be real and symmetric

• rijk also must be real and symmetric in the first index pair

• If crystal has an inversion center, then space inversionbrings one minus sign per index

• hence rijk must vanish

• only crystals without inversion center can show the

Pockels effect

• lithium niobate with 3m symmetry is an example

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Non-centrosymmetric crystals

• ε(ω,E)ij must be real and symmetric

• rijk also must be real and symmetric in the first index pair

• If crystal has an inversion center, then space inversionbrings one minus sign per index

• hence rijk must vanish

• only crystals without inversion center can show the

Pockels effect

• lithium niobate with 3m symmetry is an example

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Non-centrosymmetric crystals

• ε(ω,E)ij must be real and symmetric

• rijk also must be real and symmetric in the first index pair

• If crystal has an inversion center, then space inversionbrings one minus sign per index

• hence rijk must vanish

• only crystals without inversion center can show the

Pockels effect

• lithium niobate with 3m symmetry is an example

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Non-centrosymmetric crystals

• ε(ω,E)ij must be real and symmetric

• rijk also must be real and symmetric in the first index pair

• If crystal has an inversion center, then space inversionbrings one minus sign per index

• hence rijk must vanish

• only crystals without inversion center can show the

Pockels effect

• lithium niobate with 3m symmetry is an example

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices u = x

y

v

w

c

3m symmetry. There is a mirror plane (m) spanned by c and yand a three-fold (3) rotation symmetry u→ v → w → uaround the crystallographic axis c. c→ −c is not allowed.

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Invariant tensors

• there are four invariant tensors

• D(1) = c⊗ c⊗ c

• read D(1)ijk = cicj ck

• u⊗ u⊗ c + . . . can be simplified to

• D(2) = (x⊗ x + y ⊗ y)⊗ c

• etc.

• the Pockels tensor is a linear combination of these fourinvariant tensors

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Invariant tensors

• there are four invariant tensors

• D(1) = c⊗ c⊗ c

• read D(1)ijk = cicj ck

• u⊗ u⊗ c + . . . can be simplified to

• D(2) = (x⊗ x + y ⊗ y)⊗ c

• etc.

• the Pockels tensor is a linear combination of these fourinvariant tensors

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Invariant tensors

• there are four invariant tensors

• D(1) = c⊗ c⊗ c

• read D(1)ijk = cicj ck

• u⊗ u⊗ c + . . . can be simplified to

• D(2) = (x⊗ x + y ⊗ y)⊗ c

• etc.

• the Pockels tensor is a linear combination of these fourinvariant tensors

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Invariant tensors

• there are four invariant tensors

• D(1) = c⊗ c⊗ c

• read D(1)ijk = cicj ck

• u⊗ u⊗ c + . . . can be simplified to

• D(2) = (x⊗ x + y ⊗ y)⊗ c

• etc.

• the Pockels tensor is a linear combination of these fourinvariant tensors

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Invariant tensors

• there are four invariant tensors

• D(1) = c⊗ c⊗ c

• read D(1)ijk = cicj ck

• u⊗ u⊗ c + . . . can be simplified to

• D(2) = (x⊗ x + y ⊗ y)⊗ c

• etc.

• the Pockels tensor is a linear combination of these fourinvariant tensors

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Invariant tensors

• there are four invariant tensors

• D(1) = c⊗ c⊗ c

• read D(1)ijk = cicj ck

• u⊗ u⊗ c + . . . can be simplified to

• D(2) = (x⊗ x + y ⊗ y)⊗ c

• etc.

• the Pockels tensor is a linear combination of these fourinvariant tensors

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Invariant tensors

• there are four invariant tensors

• D(1) = c⊗ c⊗ c

• read D(1)ijk = cicj ck

• u⊗ u⊗ c + . . . can be simplified to

• D(2) = (x⊗ x + y ⊗ y)⊗ c

• etc.

• the Pockels tensor is a linear combination of these fourinvariant tensors

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Invariant tensors

• there are four invariant tensors

• D(1) = c⊗ c⊗ c

• read D(1)ijk = cicj ck

• u⊗ u⊗ c + . . . can be simplified to

• D(2) = (x⊗ x + y ⊗ y)⊗ c

• etc.

• the Pockels tensor is a linear combination of these fourinvariant tensors

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Elextric field parallel to optical axis

• assum E = E c•

ε =

n2o − n40r113E 0 00 n2o − n4or113E 00 0 n2e − n4er333E

• crystal remains birefringent

• no(E) = no − n3or113E/2• ne(E) = ne − n3er333E/2• birefringence ∆n = no − ne varies withe E• Pockels cell

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Elextric field parallel to optical axis

• assum E = E c

•

ε =

n2o − n40r113E 0 00 n2o − n4or113E 00 0 n2e − n4er333E

• crystal remains birefringent

• no(E) = no − n3or113E/2• ne(E) = ne − n3er333E/2• birefringence ∆n = no − ne varies withe E• Pockels cell

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Elextric field parallel to optical axis

• assum E = E c•

ε =

n2o − n40r113E 0 00 n2o − n4or113E 00 0 n2e − n4er333E

• crystal remains birefringent

• no(E) = no − n3or113E/2• ne(E) = ne − n3er333E/2• birefringence ∆n = no − ne varies withe E• Pockels cell

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Elextric field parallel to optical axis

• assum E = E c•

ε =

n2o − n40r113E 0 00 n2o − n4or113E 00 0 n2e − n4er333E

• crystal remains birefringent

• no(E) = no − n3or113E/2• ne(E) = ne − n3er333E/2• birefringence ∆n = no − ne varies withe E• Pockels cell

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Elextric field parallel to optical axis

• assum E = E c•

ε =

n2o − n40r113E 0 00 n2o − n4or113E 00 0 n2e − n4er333E

• crystal remains birefringent

• no(E) = no − n3or113E/2

• ne(E) = ne − n3er333E/2• birefringence ∆n = no − ne varies withe E• Pockels cell

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Elextric field parallel to optical axis

• assum E = E c•

ε =

n2o − n40r113E 0 00 n2o − n4or113E 00 0 n2e − n4er333E

• crystal remains birefringent

• no(E) = no − n3or113E/2• ne(E) = ne − n3er333E/2

• birefringence ∆n = no − ne varies withe E• Pockels cell

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Elextric field parallel to optical axis

• assum E = E c•

ε =

n2o − n40r113E 0 00 n2o − n4or113E 00 0 n2e − n4er333E

• crystal remains birefringent

• no(E) = no − n3or113E/2• ne(E) = ne − n3er333E/2• birefringence ∆n = no − ne varies withe E

• Pockels cell

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Elextric field parallel to optical axis

• assum E = E c•

ε =

n2o − n40r113E 0 00 n2o − n4or113E 00 0 n2e − n4er333E

• crystal remains birefringent

• no(E) = no − n3or113E/2• ne(E) = ne − n3er333E/2• birefringence ∆n = no − ne varies withe E• Pockels cell

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

A Pockels cell with transversal field. It may modulate or switchlight in picoseconds.

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

A Pockels cell with longitudinal field. It requires transparentelectrodes.

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

Integrated modulator with SiC

Pockels effect

Peter Hertel

Overview

Externalelectric field

Lithiumniobate

Birefringencecontrol

Devices

An integrated Mach-Zehnder interferometer